October 15, 1998 / Vol. 23, No. 20 / OPTICS LETTERS
1579
Circular solitons do not exist in photorefractive media
M. Saffman
Optics and F luid Dynamics Department, Risø National Laboratory, Postbox 49, DK-4000 Roskilde, Denmark
A. A. Zozulya
Department of Physics, Worcester Polytechnic Institute, 100 Institute Road, Worcester, Massachusetts 01609-2280
Received March 26, 1998
The shape of two-dimensional solitary beams propagating in photorefractive media with an externally applied
f ield is studied. The analytical results indicate that, for both focusing and defocusing nonlinearities, radially
symmetric self-channeled beams do not exist. Some recent experiments are interpreted in light of the present
results.  1998 Optical Society of America
OCIS codes: 160.5320, 190.5330.
The current interest in the properties of photorefractive spatial solitons was initiated by the suggestion
of Segev et al. in 1992 (Ref. 1) that self-channeled,
spatially localized beams could be observed in photorefractive media in the presence of a strong, externally
applied electric field. As was first pointed out in
Ref. 2 and subsequently demonstrated by several
groups,3,4 in the case of a single transverse coordinate hs1 1 1-dimensional fs1 1 1dDg propagationj the
photorefractive response is formally equivalent to a
saturable Kerr nonlinearity5: dn , jBj2 ys1 1 jBj2 d,
where dn is the nonlinear increment to the refractive index and B is the slowly varying envelope
of the optical field. The s1 1 1dD spatial solitons
have been observed for both focusing and defocusing
nonlinearities.2,6
By contrast, the physics of three-dimensional hs2 1
1d-dimensional fs2 1 1dDgj beam propagation in photorefractive media are remarkably different from those
observed in Kerr or saturable Kerr media that are
describable by an isotropic and local nonlinearity.
Analytical, numerical, and experimental studies4,7 – 13
have revealed phenomena that are due to the strong
anisotropy of the photorefractive nonlinearity in the
plane transverse to the direction of propagation. In
particular, an analytical solution describing a highly
elliptical self-channeled beam with characteristic diameter along the y coordinate (perpendicular to the
applied field) approximately 1.5 times larger than
along the x coordinate (parallel to the applied field)
was obtained in the limit of weak saturation of the nonlinearity.10 Numerical calculations showed that
s2 1 1dD solitary solutions are strongly elliptical over a
wide range of intensities, including the limit of strong
saturation, and convergence of a circular input beam
to an elliptical prof ile was demonstrated experimentally.10 This elliptical soliton is the analog of the
bright circular soliton that exists in s2 1 1dD saturable
Kerr-type media. Conversely, for a defocusing nonlinearity, numerical and experimental studies have
indicated that localized vortex solitons do not exist in
photorefractive media.12 Concurrently, there have
been a number of experimental reports of radially symmetric bright14 and dark15,16 solitons in photorefractive
media. Furthermore, a recent analysis17 of the same
0146-9592/98/201579-03$15.00/0
three-dimensional model as that used below [see
Eqs. (1)] appeared to demonstrate circular solitary
solutions.
The question of the spatial prof ile of solitary
solutions is of central importance in the study of threedimensional propagation effects, and the possible
existence of structurally simple, circular s2 1 1dD solitary solutions in photorefractive media is considered
here. It was shown in Ref. 4 that time-independent
three-dimensional propagation in photorefractive
media is described by the equations
µ
∂
≠
≠f
i
2 =2 Bsrd ­ is
Bsrd ,
(1a)
≠z
2
≠x
≠
(1b)
lns1 1 jBj2 d .
=2 f 1 =f ? = lns1jBj2 d ­
≠x
Equation (1a) is the parabolic wave equation in the
presence of a nonlinear increment to the refractive index proportional to ≠fy≠x. Equation (1b) determines
the form of the potential f induced by the optical beam
in the presence of an external voltage applied along the
x axis or of a photogalvanic current. This form follows
immediately from Eq. (3) of Ref. 4 in the diffusionless
approximation,18 where kDebye ! `. Depending on the
polarity of the external voltage, the factor s is equal
to 11 or 21, corresponding to a focusing or a defocusing nonlinearity, respectively. The transverse gradient operator is = ­ x̂s≠y≠xd 1 ŷs≠y≠yd, and f satisf ies
the boundary conditions =fsr' ! `d ! 0. We show
below that the anisotropy of Eqs. (1) is unavoidable in
the sense that circular solitary solutions do not exist
for any parameter values.
We seek a general soliton solution in the form
Bsx, y, zd ­ bsrdexpfiGz 1 icsx, ydg ,
(2)
p
where r ­ x2 1 y 2 . Note that the above soliton
ansatz has a radially symmetric intensity profile, but
no assumptions are made about its phase distribution.
Separating real and imaginary parts of Eq. (1a)
results in the set of equations
b=2 c 1 2s=cd ? s=bd ­ 0 ,
Gb 1
1
≠f
f2=2 b 1 s=cd2 bg ­ s
b.
2
≠x
 1998 Optical Society of America
(3a)
(3b)
1580
OPTICS LETTERS / Vol. 23, No. 20 / October 15, 1998
For s ­ 11 we seek localized bright soliton solutions satisfying limr!` bsrd ­ 0. It then follows from
Eq. (3b) that the propagation constant G . 0 and that
the soliton amplitude b has exponentially decaying
p
asymptotics at infinity, bsr ! `d ~ exps2 2G rd.
For s ­ 21 we seek dark soliton solutions with
limr!` bsrd ­ bmax , which leads to G ­ 0.
Owing to radial symmetry of the soliton amplitude
bsrd, Eq. (1b) in cylindrical coordinates sr, ud takes the
form
=2 f 1
≠f d
d
lns1 1 b2 d ­ cossud
lns1 1 b2 d .
≠r dr
dr
(4)
The solution of Eq. (4) has the form f ­ cossudhsrd,
where h satisf ies the equation
∂
µ
d2 h
h
d
1 dh
dh
2 2 1
21
lns1 1 b2 d ­ 0 .
1
2
dr
r dr
r
dr
dr
(5)
The nonlinear refractive index ≠fy≠x has the form
µ
∂
µ
∂
≠f
1 dh
h
1 dh
h
­
1
1
2
coss2ud .
(6)
≠x
2 dr
r
2 dr
r
The function h in Eq. (5) has asymptotic behavior
hsr ! `d ~ 1yr, so the nonlinear refractive index ≠fy≠x
has asymptotic form ≠fy≠x ~ 2coss2udyr 2 .
From Eq. (3b) we have
s=cd2 ­ asrd 1 ss≠fy≠xd ,
(7)
so the angular dependence of c must cancel the term
proportional to coss2ud in Eq. (6) to yield a solution.
The phase c is determined modulos2pd, so the
general solution of Eq. (3a) can be written as
X
Fl srdcosslu 1 ul d ,
(8)
c ­ mu 1
l.0
where m is an integer and the function Fl satisf ies the
equation
d2 Fl
l2
1 dFl
dFl d
2 2 Fl 1 2
ln b ­ 0 .
1
2
dr
r dr
r
dr dr
(9)
The term with l ­ 0 in Eq. (9) has been discarded, since
Z r dr̃
b22 1 const .
F0 srd ­
(10)
r̃
is divergent for both bright and vortex solitons. In
the first case limr!` bsrd ­ 0, and in the second case
bsr ­ 0d ­ 0 and limr!` bsrd ­ bmax .
Now consider the case of vortex solitons with topological charge m fi 0 [see Eq. (8)]. Asymptotics of two
linearly independent solutions for Fl are r 6l for large
arguments. Finiteness of the function Fl implies the
r 2l asymptotics at infinity. At large distances the
leading contribution to the phase gradient =c comes
from the mu term in Eq. (8) and has the form =c ­
ef myr, where ef is the azimuthal unit vector. All the
other contributions have at least 1yr 2 dependence on
the radial coordinate. This means that s=cd2 ~ m2 yr 2 .
This contribution cannot cancel the angle-dependent
term ≠fy≠x on the right-hand side of Eq. (3b), and this
concludes the proof.
For bright solitons asymptotics of two linearly independent solutions for Fl for small parguments are
r 6l and
p for large arguments are exps2 2G rd and s1 1
l2 y2 2G rd. Exponentially growing solutions do
not satisfy
p Eq. (3b), so we have to require the
s1 1 l2 y2 2G rd asymptotics for r ! `. For such
asymptotics the main contribution to the phase gradient =c comes from the azimuthal derivative; the
radial derivative is 1yr times smaller. Consequently,
at large distances s=cd2 ­ r 22 s≠cy≠ud2 . Equation (7)
in this limit yields s≠cy≠ud2 ~ bsrd 2 coss2ud. Since
s=cd2 $ 0, we should require that b $ 1. If b . 1,
then cs2pd fi cs0d, which contradicts the periodicity of
the phase with respect to the azimuthal angle u in the
absence of topological charge. Hence b ­ 1, which results immediately in c ~ cossud. This shows that only
the l ­ 1 term in Eq. (8) can be nonzero. The angledependent part of Eq. (7) then yields
µ
dF1
dr
∂2
µ
2
F1
r
∂2
­
h
dh
2
dr
r
(11)
for all r. It is easy to show that Eq. (11) cannot be
satisf ied for all r. Indeed, for large r the left-hand
side is proportional to 1yr 2 1 Os1yr 3 d (see asymptotics
for F1 ), whereas the right-hand side is proportional to
1yr 2 plus exponentially small corrections [see Eq. (5)].
This observation concludes the proof.
The above proof is valid for any relative intensity of
the light beam. Thus, in particular, in the weak saturation limit sb2 ,, 1d circular solitons are not possible,
contrary to the claims of Ref. 17. Equations (1) also
exclude the possibility of the existence of almost circular solitons. Direct substitution of any circularly symmetric envelope into Eq. (1) shows that the anisotropic
part of the nonlinear response is of the same order
of magnitude as the isotropic part. Since there is no
smallness parameter, a soliton beam (if it exists) must
be considerably different from a circular one.
These results disagree with the experimental reports given in Ref. 14 of circular solitons for s ­ 1
and in Refs. 15 and 16 for s ­ 21. We have previously explained the apparent observations of bright
circular solitons in terms of oscillatory self-focusing
in the regime of high saturation sb2 .. 1d.10,11 For
s ­ 21 previous numerical analysis of unit-charged
vortex beams shows that the vortex core rotates
and stretches on propagation, leading to complete
delocalization of the core region.12 Together with
the stretching, which occurs along a direction at a
small angle to the x coordinate, self-focusing and
compression of the core region can be observed along
the y coordinate. These effects are illustrated in
Fig. 1, where we show the evolution of the full width
at half-maximum of the core region calculated numerically from Eqs. (1) for an input field of the form
2
dp g,
B ­ expsiud f1 2 exps2rywcore dgexpf2sr 2 ywouter
with p ­ 7. wcore was chosen to give an input
core diameter of 26 mm (FWHM), while wouter was
chosen to be suff iciently large that the numerical
October 15, 1998 / Vol. 23, No. 20 / OPTICS LETTERS
Fig. 1. Evolution of the core of a unit-charged vortex.
The diameters were measured along the x (solid curve)
and the y (dotted curve) coordinates. The insets show the
intensity profiles along x and y in the central portion of the
beam s20.3 mm , x, y , 0.3 mmd.
results were unaffected by spreading of the outer
core sFWHMouter , 2 mmd. The numerical parameters were chosen to correspond to a strontium barium
niobate crystal with index of refraction n ­ 2.3, electrooptic coeff icient r33 ­ 340 pmyV, an applied field of
850 V ycm, and normalized intensity jBj2max ­ 0.95.
These values are comparable with those reported in
Ref. 16. Note that, while the core diameter measured
along y is roughly constant, the diameter along x diverges. Additional numerical studies with both larger
and smaller input beams show that the diameter along
x always diverges.
Our analysis and numerical results concerning
the propagation of vortex beams are at odds with
experimental results given in Refs. 15 and 16 of
circular photorefractive vortex solitons. To understand the apparent discrepancy one should recall
that the numerical results given in Fig. 1 were obtained with a beam having an idealized, smooth
background intensity. By contrast, a striking
feature of the intensity prof iles shown in Ref. 16
is that the background beam is strongly modulated with a peak-to-peak amplitude that exceeds
50% of the mean background level. Self-focusing
of the vortex core along both x and y coordinates
over a finite propagation distance can be observed
with nonideal, modulated background prof iles. The
asymptotic behavior is, however, always characterized
by spreading of the core. Strong background modulation also results in the fact that the input and the
output prof iles given in Fig. 1 of Ref. 16 coincide in
the center of the core but differ by 50% or more at
radial distances comparable with the core diameter.
In our opinion, it is not possible to infer the existence of
solitary solutions under such conditions.
This work was supported by NATO collaborative research grant 970173. The work of M. Saffman was
supported by grants 9502764 and 9600852 from the
Danish Natural Science Research Council.
References
1. M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys.
Rev. Lett. 68, 923 (1992).
1581
2. M. D. Iturbe-Castillo, P. A. Márquez Aguilar, J. J.
Sánchez-Mondragón, S. I. Stepanov, and V. Vysloukh,
Appl. Phys. Lett. 64, 408 (1994).
3. M. Segev, G. C. Valley, B. Crosignani, P. Di Porto,
and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994); A. A.
Zozulya and D. Z. Anderson, Opt. Lett. 20, 837 (1995);
P. A. Márquez Aguilar, J. J. Sánchez-Mondragón,
S. Stepanov, and G. Bloch, Opt. Commun. 118, 165
(1995); S. R. Singh and D. N. Christodoulides, Opt.
Commun. 118, 569 (1995); G. Montemezzani and P.
Günter, Opt. Lett. 22, 451 (1997).
4. A. A. Zozulya and D. Z. Anderson, Phys. Rev. A 51,
1520 (1995).
5. S. Gatz and J. Herrmann, J. Opt. Soc. Am. B 8, 2296
(1991).
6. G. Duree, G. Salamo, M. Segev, A. Yariv, B. Crosignani,
P. Di Porto, and E. Sharp, Opt. Lett. 19, 1195 (1994); G.
Duree, M. Morin, G. Salamo, M. Segev, B. Crosignani,
P. Di Porto, E. Sharp, and A. Yariv, Phys. Rev. Lett.
74, 1978 (1995); M. Taya, M. C. Bashaw, M. M. Fejer,
M. Segev, and G. C. Valley, Phys. Rev. A 52, 3095
(1995); M. D. Iturbe-Castillo, J. J. Sánchez-Mondragón,
S. I. Stepanov, M. B. Klein, and B. A. Weschler, Opt.
Commun. 118, 515 (1995); K. Kos, H. Meng, G. Salamo,
M.-F. Shih, M. Segev, and G. C. Valley, Phys. Rev. E
53, R4330 (1996).
7. N. Korneev, P. A. Márquez Aguilar, J. J. SánchezMondragón, S. Stepanov, M. Klein, and B. Wechsler,
J. Mod. Opt. 43, 311 (1996); G. S. Garcı́a Quirino, M. D.
Iturbe-Castillo, J. J. Sánchez-Mondragón, S. Stepanov,
and V. Vysloukh, Opt. Commun. 123, 597 (1996); C. M.
Goméz Sarabia, P. A. Márquez Aguilar, J. J. SánchezMondragón, S. Stepanov, and V. Vysloukh, J. Opt. Soc.
Am. B 13, 2767 (1996).
8. A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya,
M. S. Soskin, and M. V. Vasnetsov, Appl. Phys. B 62,
465 (1996).
9. A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A.
Zozulya, Phys. Rev. A 54, 870 (1996).
10. A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M.
Saffman, Europhys. Lett. 36, 419 (1996).
11. A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M.
Saffman, Phys. Rev. A 57, 522 (1998).
12. A. V. Mamaev, M. Saffman, and A. A. Zozulya, Phys.
Rev. Lett. 77, 4544 (1996); Phys. Rev. A 56, R1713
(1997).
13. W. Królikowski, M. Saffman, B. Luther-Davies, and C.
Denz, Phys. Rev. Lett. 80, 3240 (1998).
14. M.-F. Shih, M. Segev, G. C. Valley, G. Salamo, B.
Crosignani, and P. Di Porto, Electron. Lett. 31, 826
(1995); M.-F. Shih, P. Leach, M. Segev, M. H. Garrett,
G. Salamo, and G. C. Valley, Opt. Lett. 21, 324 (1996).
15. Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and
P. D. Maker, Phys. Rev. Lett. 78, 2948 (1997).
16. Z. Chen, M.-F. Shih, M. Segev, D. W. Wilson, R. E.
Muller, and P. D. Maker, Opt. Lett. 22, 1751 (1997).
17. B. Crosignani, P. Di Porto, A. Degasperis, M. Segev,
and S. Trillo, J. Opt. Soc. Am. B 14, 3078 (1997).
18. Corrections due to finite Debye wave-number effects
lead to additional terms in Eq. (1b) that cause bending
of solitary solutions along the x axis, as do corrections
to the denominators of the terms on the left-hand side
of Eq. (1b) (see Refs. 4 and 11). For the parameters
corresponding to experimental studies of solitary waves
in photorefractive media, these additional terms are
small and are neglected here.